MECHANISMSMachine Structure – Kinematic link, pair and chain – Grueblers criteria – Constrained motion– Degrees of freedom - Slider crank and crank rocker mechanisms – Inversions – Applications –

INTRODUCTION:

DEFINITION

The subject Theory of Machines may be defined as that branch ofEngineering-science, which deals with the study of relative motion between the various parts of amachine, and forces which act on them. The knowledge of this subject is very essential for anengineer in designing the various parts of machine.The Theory of Machines may be sub-divided into the following four branches :

1. Kinematics. It is that branch of Theory of Machines which deals with the relative motion betweenthevarious parts of the machines’

2. Dynamics. It is that branch of Theory of Machines which deals with the forces and their effects,while actingupon the machine parts in motion.

3. Kinetics. It is that branch of Theory of Machines which deals with the inertia forces which arisefrom the combined effect of the mass and motion of the machine parts.

4. Statics. It is that branch of Theory of Machines which deals with the forces and their effects whilethe ma chine parts are at rest. The mass of the parts is assumed to be negligible.

Machine:A machine consists of a number of parts or bodies we shall study the mechanisms of the

various parts or bodies from which the machine is assembled. This is done by making one of the partsas fixed, and the relative motion of other parts is determined with respect to the fixed part. Structure: It is an assemblage of a number of resistant bodies (known as members) having no relativemotion between them and meant for carrying loads having straining action. A railway bridge, a rooftruss, machine frames etc., are the examples of a structure.Kinematic link:

Each part of a machine, which moves relative to some other part, is known as akinematic link(or simply link) or element. A link may consist of several parts, which are rigidly fastened together,so that they do not move relative to one another. For example, in a reciprocating steam engine piston,piston rod and crosshead constitute one link ; connecting rod with big and small end bearings

Kinematic analysis of simple mechanisms n.

constitute a second link ; crank, crank shaft and flywheel a third link and the cylinder, engine frameand main bearings a fourth link.

A link or element need not to be a rigid body, but it must be a resistant body. A body is said to be aresistant body if it is capable of transmitting the required forces with negligible deformation. Thus alink should have the following two characteristics: 1. It should have relative motion, and 2. It must be a resistant body.

Types of Links: In order to transmit motion, the driver and the follower may be connected by the followingthree types of links:

1. Rigid link. A rigid link is one which does not undergo any deformation while transmitting motion.Strictly speaking, rigid links do not exist. However, as the deformation of a connecting rod, crank etc.of a reciprocating steam engine is not appreciable, they can be considered as rigid links. 2. Flexible link. A flexible link is one which is partly deformed in a manner not to affectthe transmission of motion. For example, belts, ropes, chains and wires are flexible links and transmittensile forces only. 3. Fluid link. A fluid link is one which is formed by having a fluid in a receptacle and themotion is transmitted through the fluid by pressure or compression only, as in the case of hydraulicpresses, jacks and brakes. kinematic Pair: The two links or elements of a machine, when in contact with each other, are said to forma pair. If the relative motion between them is completely or successfully constrained (i.e. in a definitedirection), the pair is known as kinematic pair. Let us discuss the various types of constrainedmotions.Classification of Kinematic Pairs:The kinematic pairs may be classified according to the following considerations:

1. According to the type of relative motion between the elements.

The kinematic pairs according to type of relative motion between the elements may beclassified as discussed below:(a) Sliding pair. When the two elements of a pair are connected in such a way that one can only sliderelative to the other, the pair is known as a sliding pair. The piston and cylinder, cross-head andguides of a reciprocating steam engine, ram and its guides in shaper, tail stock on the lathe bed etc.are the examples of a sliding pair. A little consideration will show that a sliding pair has a completelyconstrained motion.(b) Turning pair. When the two elements of a pair are connected in such a way that one can onlyturn or revolve about a fixed axis of another link, the pair is known as turning pair. A shaft withcollars at both ends fitted into a circular hole, the crankshaft in a journal bearing in an engine, lathespindle supported in head stock, cycle wheels turning over their axles etc. are the examples of aturning pair. A turning pair also has a completely constrained motion.(c) Rolling pair. When the two elements of a pair are connected in such a way that one rolls overanother fixed link, the pair is known as rolling pair. Ball and roller bearings are examples of rollingpair.(d) Screw pair. When the two elements of a pair are connected in such a way that one element canturn about the other by screw threads, the pair is known as screw pair. The lead screw of a lathe withnut, and bolt with a nut are examples of a screw pair.(e) Spherical pair. When the two elements of a pair are connected in such a way that one element(with spherical shape) turns or swivels about the other fixed element, the pair formed is called aspherical pair. The ball and socket joint, attachment of a car mirror, pen stand etc., are the examplesof a spherical pair.2. According to the type of contact between the elements. The kinematic pairs according to the typeof contact between the elements may be classified as discussed below:(a) Lower pair. When the two elements of a pair have a surface contact when relative motion takesplace and the surface of one element slides over the surface of the other, the pair formed is known aslower pair. It will be seen that sliding pairs, turning pairs and screw pairs form lower pairs.(b) Higher pair. When the two elements of a pair have a line or point contact when relative motiontakes place and the motion between the two elements is partly turning and partly sliding, then the pairis known as higher pair. Pair of friction discs, toothed gearing, belt and rope drives, ball and rollerbearings and cam and follower are the examples of higher pairs.

3. According to the type of closure. The kinematic pairs according to the type of closure between theelements may be classified as discussed below:(a) Self closed pair. When the two elements of a pair are connected together mechanically in such away that only required kind of relative motion occurs, it is then known as self closed pair. The lowerpairs are self closed pair.(b) Force - closed pair. When the two elements of a pair are not connected mechanically but are keptin contact by the action of external forces, the pair is said to be a force-closed pair. The cam andfollower is an example of force closed pair, as it is kept in contact by the forces exerted by spring andgravity.Kinematic Chain: When the kinematic pairs are coupled in such a way that the last link is joined to the first

link to transmit definite motion (i.e. completely or successfully constrained motion), it is called akinematic chain. In other words, a kinematic chain may be defined as a combination of kinematicpairs, joined in such a way that each link forms a part of two pairs and the relative motion betweenthe links or elements is completely or successfully constrained. For example, the crank- shaft of anengine forms a kinematic pair with the bearings which are fixed in a pair, the connecting rod with thecrank forms a second kinematic pair, the piston with the connecting rod forms a third pair and thepiston with the cylinder forms a fourth pair. The total combination of these links is a kinematic chain. If each link is assumed to form two pairs with two adjacent links, then the relation betweenthe number of pairs ( p ) forming a kinematic chain and the number of links ( l ) may be expressed inthe form of an equation :

l = 2 p – 4. . . (i)

Since in a kinematic chain each link forms a part of two pairs, therefore there will be as many linksas the number of pairs. Another relation between the number of links (l) and the number of joints ( j ) whichconstitute a kinematic chain is given by the expression :

The equations (i) and (ii) are applicable only to kinematic chains, in which lower pairs are used.These equations may also be applied to kinematic chains, in which higher pairs are used. In thatcase each higher pair may be taken as equivalent to two lower pairs with an additional element orlink.

Let us apply the above equations to the following cases to determine whether each ofthem is a kinematic chain or not.

1. Consider the arrangement of three links A B, BC and C A with pin joints aA,B and C.

Number of links, l = 3

Number of pairs, p=3

and number of joints, j = 3

From equation (i), l=2p-3

3=2*3-4=2

L.H.S. > R.H.SNow from equation (ii),`

i.e. L.H.S. > R.H.S

Mechanism When one of the links of a kinematic chain is fixed, the chain is known as mechanism. Itmay be used for transmitting or transforming motion e.g. engine indicators, typewriter etc A mechanism with four links is known as simple mechanism, and the mechanism withmore than four links is known as compound mechanism. When a mechanism is required totransmit power or to do some particular type of work, it then becomes a machine. In such cases, thevarious links or elements have to be designed to withstand the forces (both static and kinetic) safely.

A little consideration will show that a mechanism may be regarded as a machine in which each partis reduced to the simplest form to transmit the required motion.Forces Acting in a MechanismConsider a mechanism of a four bar chain, as shown in Fig. Let force FA newton is acting at thejoint A in thedirection of the velocity of A (vAm/s) which is perpendicular to the link D A. Suppose a force FBnewton is transmitted to the joint B in the direction of the velocity of B (i.e. vB m/s) which is

perpendicular to the link CB. If we neglectthe effect of friction and the change of kinetic energy of the link (i.e.), assuming the efficiency oftransmission as 100%), then by the principle of conservation of energy,

Types of Kinematic Chains

The most important kinematic chains are those which consist of four lower pairs, each pairbeing a sliding pair or a turning pair. The following three types of kinematic chains with four lowerpairs are important from the subject point of view

1. Four bar chain or quadric cyclic chain,2. Single slider crank chain, and3. Double slider crank chain.

These kinematic chains are discussed, in detail, in the following articles

Four Bar Chain or Quadric Cycle ChainWe have already discussed that the kinematic chain is a combination of four or more

kinematic pairs, such that the relative motion between the links or elements is completelyconstrained. The simplest and the basic kinematic chain is a four bar chain or quadric cycle chain. Itconsists of four links, each of them forms a turning pair at A, B, C and D. The four links may be ofdifferent lengths. According to Grashof ’s law for a four bar mechanism, the sum of the shortestand longest link lengths should not be greater than the sum of the remaining two link lengths ifthere is to be continuous relative motion between the two links. A very important consideration in designing a mechanism is to ensure that the inputcrank makes a complete revolution relative to the other links. The mechanism in which no linkmakes a complete revolution will not be useful. In a four bar chain, one of the links, in particular theshortest link, will make a complete revolution relative to the other three links, if it satisfies theGrashof ’s law. Such a link is known as crank or driver. A D (link 4) is a crank. The link BC (link2) which makes a partial rotation or oscillates is known as lever or rocker or follower and the linkCD (link 3) which connects the crank and lever is called connecting rod or coupler. The fixed linkA B (link 1) is known as frame of the mechanism. When the crank (link 4) is the driver, themechanism is transforming rotary motion into oscillating motion.

Single Slider Crank Chain A single slider crank chain is a modification of the basic four bar chain. It consist of one slidingpair and three turning pairs. It is, usually, found in reciprocating steam engine mechanism. This typeof mechanism converts rotary motion into reciprocating motion and vice versa. In a single slidercrank chain, as shown the links 1 and 2, links 2 and 3, and links 3 and 4 form three turning pairswhile the links 4 and 1 form a sliding pair. The link 1 corresponds to the frame of the engine, which is fixed. The link 2 corresponds tothe crank ; link 3 corresponds to the connecting rod and link 4 corresponds to cross-head. As thecrank rotates, the cross-head reciprocates in the guides and thus the piston reciprocates in thecylinder.Double Slider Crank Chain A kinematic chain which consists of two turning pairs and two sliding pairs is known asdouble slider crank chain. We see that the link 2 and link 1 form one turning pair and link 2 andlink 3 form the second turning pair. The link 3 and link 4 form one sliding pair and link 1 and link 4form the second sliding pair.

Grubler’s Criterion for Plane Mechanisms: The Grubler’s criterion applies to mechanisms with only single degree of freedom jointswhere the overall movability of the mechanism is unity. Substituting n = 1 and h = 0 in Kutzbachequation, we have 1 = 3 (l – 1) – 2 j or 3l – 2j – 4 = 0This equation is known as the Grubler's criterion for plane mechanisms with constrained motion.A little consideration will show that a plane mechanism with a movability of 1 and only singledegree of freedom joints cannot have odd number of links. The simplest possible mechanisms ofthis type are a four bar mechanism and a slider-crank mechanism in which l = 4 and j = 4

Types of Constrained Motions: Following are the three types of constrained motions: 1. Completely constrained motion. When the motion between a pair is limited to adefinite direction irrespective of the direction of force applied, then the motion is said to be acompletely constrained motion. For example, the piston and cylinder (in a steam engine) form a pairand the motion of the piston is limited to a definite direction (i.e. it will only reciprocate) relative tothe cylinder irrespective of the direction of motion of the crank. The motion of a square bar in asquare hole, and the motion of a shaft with collars at each end in a circular hole, are also examplesof completely constrained motion. 2. Incompletely constrained motion. When the motion between a pair can take place in morethan one direction, then the motion is called an incompletely constrained motion. The change in thedirection of impressed force may alter the direction of relative motion between the pair. A circularbar or shaft in a circular hole, is an example of an incompletely constrained motion as it may eitherrotate or slide in a hole. These both motions have no relationship with the other. 3. Successfully constrained motion. When the motion between the elements, forming apair, is such that the constrained motion is not completed by itself, but by some other means, then

the motion is said to be successfully constrained motion. Consider a shaft in a foot-step bearing.The shaft may rotate in a bearing or it may move upwards. This is a case of incompletely constrained motion. But if the load is placed on the shaft to prevent axial upward movement of theshaft, then the motion of the pair is said to be successfully constrained motion. The motion of anI.C. engine valve (these are kept on their spring) and the piston reciprocating inside an enginecylinder seat by a are also the examples of successfully constrained motion.Degrees of Freedom for Plane Mechanisms: In the design or analysis of a mechanism, one of the most important concern is thenumber of degrees of freedom (also called movability) of the mechanism. It is defined as thenumber of input parameters (usually pair variables) which must be independently controlled in orderto bring the mechanism into a useful engineering purpose. It is possible to determine the number ofdegrees offreedom of a mechanism directly from the number of links and the number and types of jointswhich it includes.

Consider a four bar chain, as shown in Fig., A little consideration will show that only one variablesuch as ϴ is needed to define the relative positions of all the links. In other words, we say that thenumber of degrees of freedom of a four bar chain is one. Now, let us consider a five bar chain, asshown in Fig., In this case two variables such as ϴ1 and ϴ2 are needed to define completely the

relative positions of all the links. Thus, we say that the number of degrees of freedom is * two.In order to develop the relationship in general, consider two links A B and CD in a plane motion asshown in fig i.

The link AB with co-ordinate system O X Y is taken as the reference link (or fixed link). Theposition of point P on the moving link CD can be completely specified by the three variables, i.e.the co-ordinates of the point P denoted by x and y and the inclination θ of the link CD with X-axisor link A B. In other words, we can say that each link of a mechanism has three degrees of freedombefore it is connected to any other link. But when the link CD is connected to the link A B by aturning pair at A , the position of link CD is now determined by a single variable θ and thus has onedegree of freedom.Slider crank–crank rocker mechanism: A slider –mechanism in which OA is the crank moving with uniform angular velocity in theclockwise direction. At point B, a slider moves on the fixed guide G. AB is the coupler joining A atB. It is required to find the velocity of the slider at B.Writing the velocity vector equation,Vel. of B rel. to O = vel. Of B to A + vel. Of A rel. to O Vbo= vba+vao; vbg=vao+vbaVbo is replaced by vbg as O and G are two points on fixed link with zero relative between them.

Take the vector vao which is completely known.

Vao= ὼ.OA; ┴ to OA

Vba is ┴ AB, draw a line ┴ AB through a;

Through g draw a line parallel to the motion of B. the intersection of the two lines locates the pointb. For the given configuration, the coupler AB has angular velocity in the counter- clockwisedirection, the magnitude being vba/ BA.

INVERSION: We have already discussed that when one of links is fixed in a kinematic chain, it is called amechanism. So we can obtain as many mechanisms as the number of links in a kinematic chain byfixing, in turn, different links in a kinematic chain. This method of obtaining different mechanismsby fixing different links in a kinematic chain, is known as inversion of the mechanism.It may be noted that the relative motions between the various links is not changed in any mannerthrough the process of inversion, but their absolute motions (those measured with respect to thefixed link) may be changed drastically.

APPLICATIONS:

INVERSIONS OF FOUR BAR CHAINThough there are many inversions of the four bar chain, yet the following are important from thesubject point of view :

1. Beam engine (crank and lever mechanism) A part of the mechanism of a beam engine (also known as crank and lever mechanism) whichconsists of four links . In this mechanism, when the crank rotates about the fixed centre A, the leveroscillates about a fixed centre D. The end E of the lever CDE is connected to a piston rod whichreciprocates due to the rotation of the crank. In other words, the purpose of this mechanism is toconvert rotary motion into reciprocating motion.2. Coupling rod of a locomotive (Double crank mechanism). The mechanism of a coupling rod of a locomotive (also known as double crankmechanism) which consists of four links. In this mechanism, the links AD and BC (having equallength) act as cranks and are connected to the respective wheels. The link CD acts as a coupling rodand the link A B is fixed in order to maintain a constant centre to centre distance between them. Thismechanism is meant for transmitting rotary motion from one wheel to the other wheel.

INVERSIONS OF SINGLE SLIDER CRANK CHAIN:

We have seen in the previous article that a single slider crank chain is a four-linkmechanism. We know that by fixing, in turn, different links in a kinematic chain, an inversion isobtained and we can obtain as many mechanisms as the links in a kinematic chain. It is thusobvious, that four inversions of a single slider crank chain are possible. These inversions are foundin the followingMechanisms.

Pendulum pump or Bull engine.In this mechanism, the inversion is obtained by fixing the cylinder or link 4 (i.e. sliding pair). In thiscase, when the crank (link 2) rotates, the connecting rod (link 3) oscillates about a pin pivoted to thefixed link 4 at A and the piston attached to the piston rod (link 1) reciprocates. The duplex pumpwhich is used to supply feed water to boilers have two pistons attached to link 1.Crank and slotted lever quick return motion mechanism:

This mechanism is mostly used in shaping machines, slotting machines and in rotary internalcombustion engines. this mechanism, the link AC (i.e. link 3) forming the turning pair is fixed, Thelink 3 corresponds to the connecting rod of a reciprocating steam engine. The driving crank CBrevolves with uniform angular speed about the fixed centre C. A sliding block attached to thecrankpin at B slides along the slotted bar AP and thus causes AP to oscillate about the pivoted pointA . A short link PR transmits the motion from AP to the ram which carries the tool and reciprocatesalong the line of stroke R1R2. The line of stroke of the ram (i.e. R1R2) is perpendicular to AC

produced.In the extreme positions, AP1 and AP2 are tangential to the circle and the cutting tool is at the end

of the stroke. The forward or cutting stroke occurs when the crank rotates from the position CB1 toCB2 (or through an angle â) in the clockwise direction. The return stroke occurs when the crank

rotates from the position CB2 to CB1 (or through angle á)in the clockwise direction. Since the crank has uniform angular speed.

INVERSIONS OF DOUBLE SLIDER CRANK CHAINThe following three inversions of a double slider crank chain are important from the subject point ofview :Elliptical trammels. It is an instrument used for drawing ellipses. This inversion is obtained by fixing theslotted plate (link 4),The fixed plate or link 4 has two straight grooves cut in it, at right angles toeach other. The link 1 and link 3, are known as sliders and form sliding pairs with link 4. The link AB (link 2) is a bar which forms turning pair with links 1 and 3. When the links 1 and 3 slide alongtheir respective grooves, any point on the link 2 such as P traces out an ellipse on the surface of link4, A little consideration will show that AP and BP are the semi-major axis and semi-minor axis ofthe ellipse respectively.METHODS FOR DETERMINING THE VELOCITY OF A POINT ON A LINK: Though there are many methods for determining the velocity of any point on a link in amechanism whose direction of motion (i.e. path) and velocity of some other point on the same linkis known in magnitude and direction, yet the following two methods are important from the subjectpoint of view.

1. Relative velocity method, and2. Instantaneous centre method.

Velocity in mechanisms (relative velocity method) RELATIVE VELOCITY OF TWO BODIES MOVING IN STRAIGHT LINESHere we shall discuss the application of vectors for the relative velocity of two bodies moving alongparallel linesand inclined lines, as shown in Fig. 1 (a) and 2 (a)respectively. Consider two bodies A and B moving along parallel lines in the same direction withabsolute velocities vAandvB such that vA> vB, as shown in Fig. 1 (a). The relative velocity of A with respect to B ,

From Fig.1 (b), the relative velocity of A with respect to B (i.e. vAB) may be written in the vectorform as follows:

Now consider the body B moving in an inclined direction as shown in Fig. 2 (a). The relativevelocity of A with respect to B may be obtained by the law of parallelogram of velocities or trianglelaw of velocities. Take any fixed point o and draw vector oa to represent vA in magnitude anddirection to some suitable scale. Similarly, draw vector ob to represent vB in magnitude and

direction to the same scale. Then vector ba represents the relative velocity of A with respect to B asshown in Fig. 2 (b). In the similar way as discussed above, the relative velocity of A with respect toB,

From above, we conclude that the relative velocity of point A with respect to B (vAB) and the

relative velocity of point B with respect A (vBA) are equal in magnitude but opposite in direction,i.e.

Motion of a LinkConsider two points A and B on a rigid link A B, asshown in Fig 3 (a). Let one of the extremities (B) of the linkmove relative to A , in a clockwise direction. Since the distance from A to B remains the same,therefore there can be norelative motion between A and B, along the line A B. It is thus obvious, that the relative motion of B with respect to A must be perpendicular to A B.Hence velocity of any point on a link with respect to another point on the same link is alwaysperpendicular to the line joining these points on the configuration (or space) diagram.

Thus, we see from equation (iii), that the point c on the vector ab divides it in the same ratio as Cdivides the link A B.Note: The relative velocity of A with respect to B is represented by ba, although A may be a fixedpoint. The motion between A and B is only relative. Moreover, it is immaterial whether the linkmoves about A in a clockwise direction or about B in a clockwise direction. Velocity of a Point ona Link by Relative Velocity Method The relative velocity method is based upon the relative velocityof the various points of the link as discussed in Art..3. Consider two points A and B on a link asshown in Fig..4 (a). Let the absolute velocity of the point A i.e. vA is known in magnitude and

direction and the absolute velocity of the point B i.e. vB is known in direction only. Then thevelocity of B may be determined by drawing the velocity diagram as shown in Fig. 4 (b). Thevelocity diagram is drawn as follows : 1. Take some convenient point o, known as the pole. 2. Through o, draw oa parallel and equal to vA, to some suitable scale.

3. Through a, draw a line perpendicular to A B of Fig. 4 (a). This line will represent thevelocity of B with respect to A , i.e. vBA.

4. Through o, draw a line parallel to vBintersecting the line of vBA at b.

5. Measure ob, which gives the required velocity of point B ( vB), to the scale

Velocities in Slider Crank MechanismIn the previous article, we have discused the relative velocity method for the velocity of any pointon a link, whose direction of motion and velocity of some other point on the same link is known.The same method may also be applied for the velocities in a slider crank mechanism. A slider crankmechanism is shown in Fig. 5 (a). The slider A is attached to the connecting rod A B. Let the radiusof crank OB be r and let it rotates in a clockwise direction, about the point O with uniform angularvelocity ὼ rad/s. Therefore, the velocity of B i.e. vB is known in magnitude and direction. The

slider reciprocates along the line of stroke A O. The velocity of the slider A (i.e. vA) may bedetermined by relative velocity method as discussed below :1.From any point o, draw vector ob parallel to the direction of vB (or perpendicular to OB) such

that ob = vB= ὼ.r, to some suitable scale, as shown in Fig. 5 (b).

2. Since A B is a rigid link, therefore the velocity of A relative to B is perpendicular to A B. Nowdraw vector ba perpendicular to A B to represent the velocity of A with respect to B i.e. vAB.

3. From point o, draw vector oa parallel to the path of motion of the slider A (which is along AOonly). The vectors ba and oa intersect at a. Now oa represents the velocity of the slider A i.e. vA, to

the scale. The angular velocity of the connecting rod A B (ὼAB) may be determined as follows:

The direction of vector ab (or ba) determines the sense of ὼAB which shows that it is

anticlockwise.

PROBLEMS:Example 1.In a four bar chain ABCD, AD is fixed and is 150 mm long. The crank AB is 40 mm long androtates at 120 r.p.m. clockwise, while the link CD = 80 mm oscillates about D. BC and AD are ofequal length. Find the angular velocity of link CD when angle BAD =60°.

GIVEN :NBA=120 r.p.m

ὼ=2π×120/60 =12.568 rad/sBAD =60°CD = 80 mmSOLUTION:Since the length of crank A B = 40 mm = 0.04 m, therefore velocity of B with respect to A orvelocity of B, (because A is a fixed point),

vBA= vB=ὼBA× A B = 12.568 × 0.04 = 0.503 m/sFirst of all, draw the space diagram to some suitable scale, as shown in Fig. 6 (a). Now the velocitydiagram, as shown in Fig. 6(b), is drawn as discussed below :